Lemma (Proposition 8 of [1]}) 
Consider a random vector \(Z \in \mathbb {R}^n\)  and a random index \(I \in [n]\) . Suppose for each \(i\in [n]\)  that \(Z_i\)  has mean \(\mu _i\)  and the distribution of \(Z_i - \mu _i\)  is sub-Gaussian with variance proxy \(\sigma _i^2\) . Then
\(|\mathbb {E}\left[Z_I - \mu _I \right]| \leqslant \sqrt{ \mathbb {E}\left[ \sigma _I^2 \right] } \sqrt{2 I\left( Z; I \right) }.\) 

